Figure 21: Depth after 0 s, 7 s, 14 s, 21 s, 28s, 35 s, 42 s and 50 s for the dam-break test case.
Figure 22: Depth after 50 s for the dam-break test computed with (from top to bottom) RK3 (left) and explPeer (right) with forward-backward Euler scheme, ROS3Pw, implPeer2 and implPeer3 with simplified Jacobian in cut cells only (left) and everywhere (right).
7 Conclusions
In this thesis we presented a new methodology to describe time-splitting methods for the solution of the compressible Euler equations. The basic principle of the presented approach is that we split the Euler equations into the advection part and the acoustics and make the assumption that one part, the acoustics, of the split differential equation can be solved analytically so that stability and order investigations can be made for the underlying method which solves the advection part. With this methodology we considered split-explicit Runge-Kutta methods and reproduced the known stability results.
We introduced the class of explicit peer methods and used them as the underlying method for the solution of split differential equations. We presented order conditions and stability results and finally derived the split-explicit peer method explPeer which is as accurate and efficient as the common split-explicit Runge-Kutta method RK3.
Particularly linear stability analysis showed that explPeer, even without any artificial damping, is as stable as RK3 with divergence damping. The derived split-explicit peer method explPeer also remains stable when using the trapezoidal rule as integrator for the acoustics in contrast to RK3, which is not sufficiently stable and therefore needs off-centering which results in a reduction of order down to 1.
The computational effort of explPeer is comparable to RK3: Both have s= 3 stages and therefore need three evaluations of the expensive, slow part of the right-hand side per large time step. Because the sum of the fast integration intervals of explPeer||α||1 = 1.44 is about 20% smaller than the sum of the nodes of RK3 (1/3+1/2+1=11/6) the expense for the evaluation of the fast part is a slightly lower for the peer method. In general peer methods have a higher overhead than Runge-Kutta methods because they use more linear combinations of the numerical solution and its function evaluations but for partial differential equations this fact is a negligible disadvantage because of the very expensive right-hand side.
One disadvantage of the peer method might be the memory requirement. Because only the secondary diagonal of the Butcher tableau of RK3 has non-zero entries RK3 needs the memory capacity for three numerical solutions and function evaluations: One for the initial value, another for the function evaluation and the third for the updated fast part. In contrast the peer method needs twice of this memory: s= 3 for the linear combinations eTi BYm−1 +eTi SYm which are successively updated to Ymi, i.e. in the ith stage the integration of the fast part starts with the initial value eTiBYm−1 +eTiSYm and every update is stored at the same place so that in the end eTi BYm−1+eTi SYm is overwritten by Ymi. The same holds for the function evaluations so that the memory for six solutions is needed altogether. Because in practical applications problems are solved in a massively parallel environment which usually uses domain decomposition techniques every CPU has to know only the values on its small part of the whole domain and therefore the requirement for twice of the memory capacity is no relevant disadvantage, there are enough reserves.
The split-explicit peer method explPeer with the trapezoidal rule can be interpreted as a linearly implicit peer method whose Jacobian incorporates acoustics only. Unfor-tunately this linearly implicit peer method is not much more stable when it uses the simplified Jacobian which additionally includes advection and diffusion. This fact
moti-vated us to consider linearly implicit peer methods which have the same good stability properties of explPeer when they use the partial Jacobian which only incorporates acous-tics, but furthermore they should be stable for arbitrary large CFL numbers if they use the full but simplified Jacobian. This property is necessary because we use cut cells for the representation of orography which can result in arbitrary small cells.
We derived order conditions for the class of linearly implicit peer methods which allow the construction of methods that retain the full order independently of what is used as Jacobian. Furthermore we presented a condition which is sufficient for superconvergence.
With this order theory we constructed two linearly implicit methods with three stages and order of consistency 2. They are A-stable in the common sense (not shown). The method implPeer2 was derived from the coefficients of explPeer, while implPeer3 was constructed from new coefficients in order to obtain a superconvergent third-order method. For wind speeds smaller thancs/6 they are stable for arbitrary large advection and acoustic CFL numbers if the simplified Jacobian is used. They are as stable as explPeer, i.e. stable for arbitrary large acoustic CFL numbers andCadv <1.7, if the Jacobian only incorporates the acoustic part of the compressible Euler equations. We found these methods with a genetic algorithm, where we optimized the degrees of freedom with respect to small amplitude and phase errors with the desired order conditions and stability criteria as side conditions.
The simplified Jacobian corresponds to the advection form of the Euler equation while the conservative form was used as the right-hand side to conserve mass, momentum and entropy. It uses a lower-order spatial discretization. Furthermore it is used in cut cells only while in the remaining domain the partial Jacobian, which only incorporates the acoustic part of the Euler equations, is used. Table 1 on page 55 shows the amount of memory which is needed for the Jacobian per grid cell. The partial Jacobian needs less memory than the simplified Jacobian and much less than the exact Jacobian. Because in numerical weather prediction models orography appears at ground level, cut cells can be located only there, i.e. in a model with 50 vertical layers only 2% of the cells in the whole domain are cut cells, which is negligible from the memory point of view. Therefore the amount of memory for the Jacobian in applications where the full but simplified Jacobian is used in cut cells only is nearly the same as for the partial Jacobian. Furthermore there is a theoretical speed-up of 2.3 for a matrix-vector multiplication when using the partial Jacobian instead of the simplified Jacobian. The practical speed-up might be even larger because there more entries have to be computed and the systems might be worse conditioned if the simplified Jacobian is used. In comparison to the exact Jacobian the speed-up would even be 5.7. The computing time for the zeppelin test with the simplified Jacobian for implPeer2 (without the time needed for computing the initial values) is 608 s with MATLAB on an Intel Core 2 Quad Q9550 @ 2833 Mhz with 3 GB RAM. When using the full but simplified Jacobian in cut cells only the computation takes 405 s, i.e.
33% of the computing time is saved, which corresponds to a speed-up of 1.5. The times needed for setting up and solving the linear systems of equations are 310 s respectively 113 s, i.e. the speed-up for this part of the solver is 2.7. When applying explPeer with the trapezoidal rule to the flow over a mountain test case with a time step size of 0.2 s the computation takes 13146 s (3812 s) where the time in parentheses is needed for setting up and solving the linear systems of equations. The times for implPeer2 with
the same time step size are 13251 s (3975 s) when the simplified Jacobian is used in cut cells only and 20011 s (10560 s) when it is used everywhere. So the speed-up when using the simplified Jacobian in cut cells only is 1.5 (2.6). The difference in computing time between the split-explicit method and the partially implicit method is just about 1% (4%), i.e. the partially implicit method is as efficient as the split-explicit method. As mentioned before the systems were solved with the built-in MATLAB solver which uses LU decomposition so the results may differ in 3D applications where iterative solvers are used. On the other hand the theoretical speed-up increases in higher dimensions.
Nevertheless these values give a good insight into what computational time can be saved when computing with the full but simplified Jacobian in cut cells only. Furthermore the partial and simplified Jacobians are easier to implement than the exact Jacobian due to the smaller stencils, which makes the implementation of block-structured grids and parallelization more comfortable. We gave a detailed insight into the stability analysis which incorporates the influences of these simplifications.
The application of the split-explicit methods to the compressible Euler equations in conservative flux-form with a finite volume spatial discretization confirmed the linear stability results: Both methods stably integrate the test problems. So we found a split-explicit method for the compressible equations which is as accurate, stable and efficient as RK3 but without the need for damping. The split-explicit peer method is stable even for a very high number of small time steps and therefore it is appropriate for solving low Mach number problems. The use of a time step size which nearly corresponds to the Courant number from linear stability theory still produces an acceptable solution.
The tests which incorporate orography, i.e. the flow around a mountain test, the zeppelin test and the dam-break test, all reveal the same results: No differences between the solutions computed with the simplified Jacobian in cut cells only and the simplified Jacobian everywhere are visible. Despite the large CFL numbers in cut cells of O(100) for advection respectively O(5000) for sound waves and advection CFL numbers in the free atmosphere, which are close to the stability maximum of 1.7, the solutions show no noise or instabilities, i.e. the results from linear stability theory are valid for the nonlinear Euler equations. Furthermore the peer methods with the partial and the full but simplified Jacobian harmonize very well even if the Jacobian is dynamically adapted, as in the density current test where the wind speed determines whether advection and diffusion are incorporated in the Jacobian or not. Because of the high wind speeds in the rising bubble and the density current test there are some differences visible between the solutions computed with partial and simplified Jacobians. The solutions where advection is treated explicitly are qualitatively better, i.e. not only from the efficiency point of view but also because of the accuracy the Jacobian should incorporate advection and diffusion only where it is necessary for stability. There are two reasonable applications for treating advection implicitly not only in cut cells but also in cells with high wind speeds: Firstly in numerical weather prediction models the jet stream has to be implemented which can reach velocities of more than 100 m s−1. This is much faster than the wind speeds in the remaining atmosphere, i.e. in explicit models which use no multi-rate schemes the CFL condition for the jet stream limits the maximum time step size. If the Jacobian incorporates advection in regions where the jet stream is known to be or in regions with high wind speeds the time step size is not restricted by the speed of the jet stream which
results in computational efficiency. Secondly in numerical weather prediction the used time step sizes are not close to the CFL condition because the occurring wind speeds are not known a priori. A dynamical adaption of the Jacobian in case of higher wind speeds allows larger time steps. Furthermore this strategy guarantees that the CFL condition cannot be violated, i.e. this ansatz not only results in computational efficiency but also in reliability of the model.
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Lebenslauf
Persönliche Daten
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Stefan Jebens am 17. März 1984 in Hamburg geboren
Ausbildung
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August 1990 – Juli 1992 Grundschule Moorflagen (3. Klasse übersprungen)
August 1992 – Juli 1993 Katholische Schule Hamburg August 1993 – Januar 1994 Johanneum
(humanistisches Gymnasium)
Februar 1994 – Januar 1997 Wilhelm-Gymnasium (humanistisches Gymnasium)
Februar 1997 – Juli 1999 Albert-Schweitzer-Gymnasium (allgemeines Gymnasium)
August 1999 – Juni 2002 Wirtschaftsgymnasium H7 City Nord Abschluss mit Abitur, Note 1,1
LK Mathematik: 1+
LK Wirtschaft: 1
GK Deutsch: 3+
GK Physik: 1+
Oktober 2002 – Juni 2007 Mathematik-Studium an der Universität Halle Abschluss als Diplom-Mathematiker, Note 1,4
Vertiefungsrichtung: Numerische Mathematik
Praktikumsarbeit: Parallele explizite Zweischritt-Peer-Methoden
Diplomarbeit: Explizite Mehrschritt-Peer-Methoden für spezielle Systeme 2. Ordnung
seit Juli 2007 Promotion an der Universität Halle
Berufspraxis
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August 1998 – September 1998 Praktikum bei der Lufthansa Technik AG April 2004 – Oktober 2004 Callcenter-Agent für die Tele2 GmbH
November 2004 – Juni 2007 Nachhilfelehrer beim Lernstudio Barbarossa Oktober 2005 – März 2006 Forschungspraktikum an der Universität Halle
Juli 2007 – August 2010 Wissenschaftlicher Mitarbeiter am Leibniz-Institut für Troposphärenforschung
seit August 2010 Lehrer am Schiller-Gymnasium Köln